Agenda

• Parameters vs. Estimates
• Confidence Intervals
• The Parameters in Linear Regression
• Quantifying Uncertainty in Linear Regression

From sample to population

• Often we want to know some characteristic about a population of interest
  • We call this characteristic a “parameter”
• But we only have a sample of that population
  • We call this the “estimate”
• How do we make inferences about the population parameter with what we learn from our estimate?

Empirical Example: Brexit

• Data from the British Electoral Survey

• We want to learn the probability that a Brithis citizien supports Brexiting in the country.

• Notice that such probability is the same in expectation as the proportion of pro-brexiting citizens

• We have a random sample of British citizens who were surveyed

Empirical Example: Brexit

brex <- read.csv(here::here("./slides/code/BES.csv"))
str(brex)

'data.frame':   30895 obs. of  4 variables:
 $ vote     : chr  "leave" "leave" "stay" "leave" ...
 $ leave    : int  1 1 0 1 NA 0 1 1 1 1 ...
 $ education: int  3 NA 5 4 2 4 3 2 3 4 ...
 $ age      : int  60 56 73 64 68 85 78 51 59 68 ...

table(brex$vote, useNA = "always")

   don't know         leave          stay wouldn't vote          <NA> 
         2314         13692         14352           537             0 

brex$exit <- ifelse(brex$vote=="leave", 1, 0)
prop.table(table(brex$exit))*100

       0        1 
55.68215 44.31785 

Empirical Example: Brexit

• “Support of Brexit” is a Random Variable:

  • \(Support \sim \text{Bernoulli}(p)\)

• With a probability mass function:

• Where \(E(Support) = p\) and \(Var(Support) = p(1-p)\)

Empirical Example: Brexit

• We want to know what \(p\) is!

• But it is a population parameter and we only have a sample

• We can estimate \(\hat{p}\) by doing

phat <- mean(brex$exit)
phat

[1] 0.4431785

• But if our sample were slightly different, we would have gotten a different \(\hat{p}\)

Sampling Distribution of \(\hat{p}\)

require(tidyverse)
set.seed(7)

out.means <- c()

for (i in 1:1000) {
  temp_dat <- sample_n(brex, nrow(brex), replace = T)
  out.means[i] <- mean(temp_dat$exit)
  rm(temp_dat)
}

hist(out.means)
abline(v = mean(out.means), col = "red", lwd = 2, add = T)

Sampling Distribution of \(\hat{p}\)

• That’s just a normal distribution!

• All normal distributions can be described by their mean and their standard deviaton

• This one is called “sampling distribution of the sample mean”

  • Centered around our estimate
  • \(SE = \sqrt{\frac{Var (Support)}{n}}\)

• Knowing it is a normal distribution helps us quantify the uncertainty in our estimates

Standard Errors

• The standard deviation of the sampling distribution of an estimator is called “standard error”

• One interpretation: “How off are our estimates, on average, from the true population parameter?”

• By calculating the standard error we can know the shape of the sampling distribution. This helps us do two important things:

  1. Construct confidence intervals (what is the range within which the true value is likely to be?)
  2. Do hypothesis testing (p-values and statistical significance)

Confidence Intervals

• Range of values that likely includes the true value of our parameter of interest

• Specifically, the range that includes a pre-specified proportion of the density of the sampling distribution of our sampling distribution

• Interpretation: “With X% confidence, I know my true parameter is within the confidence interval”

  • Confidence is NOT probability!

Confidence Intervals

• E.g: Because of the properties of the normal distribution, we know that 95% of the density will be within the following range:

\[\begin{align*}\small CI_{95\%} = \hat{p} - 1.96 \times \sqrt{\frac{Var (Support)}{n}},\\ \hat{p} + 1.96 \times \sqrt{\frac{Var (Support)}{n}} \end{align*}\]

• Interpretation: With 95% confidence, the true value of \(p\) is within that interval